Real Numbers In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 (1.41421356… the square root of two, an irrational algebraic number) and π (3.14159265…, a transcendental number). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Real numbers can be thought of as points on an infinitely long number line Set of all real numbers is usually denoted by = . . . . . . . . . . . . . . . . . Further symbol ref Visit: http://www.rapidtables.com/math/symbols/Basic_Math_Symbols.htm Exercise -­‐A * Express each of the following rational number as a decimal number, Identify the two different set of rational numbers. i) 6/8 ii)11/15 iii) 7/6 iv) 17/16 v) √(4/9) * Which of the flowing real numbers are irrational numbers. List them separately. i) √2 ii) √4 iii) √5 iv) √6 v) √7 vi) √8 vii) √9 viii) √10 List any four rational numbers and any four irrational numbers …………………………………………… …………………………………………… …………………………………………… …………………………………………… …………………………………………… …………………………………………… Prepared by -­‐PBP Darshana de Silva, (B.Sc, PgDBM, MBS) University of Colombo 1 Surds Numbers such as √2 whose value can’t be decided are known as surds. Since the value of a number like √9 can be decide =+3 , it is not considered as a surds. However a irrational number such as √3, √6 , √7and √11 are come under this. Some times can be denoted in simple form as a product of rational factor and an irrational factor. eg: √20 = √(4x5) = √4x √5 = 2√5 , √. . . . = √(25x9x5) = √ . . . . . Exercise -­‐B Addition & Substation The surds 7√5 and 2√5 have the same irrational factor. Surds with same irrational factor such as these are also surds. Mathematical operations used for the addition and subtraction of like algebraic terms can be used in the same manner for the like surds. Let us consider few examples in this manner. i) 7√5+ 3√5 = (7+3) √5 = 10√5 ii) 7√5-­‐ 3√5 = (7-­‐3) √5 = 4√5 iii) 4√3+ 5√3-­‐ 6√3 = . . . . . . . . . . . . . . . . . . .=. . . . . . . . . . . iv) 6√3+ 5√2-­‐ 4√3 -­‐√2 = . . . . . . . . . . . . . . . . . . .=. . . . . . . . . . . Multiplication and Division Multiplication i) 6√3 x 5√2 = 6x5x√(3x2) = 30√6 ii) 3√15x √10 = 2x√(15x10) = 2 x √(25x6) = 2√25 x √6 = 2x5x √6 =10√6 Division i) 3√8÷4√12 = 3√8⁄4√12 = 3 √(….x2)/4√ (4x…..) = (3x√… x √2)/ (4x√4x …..) = 3√2/4√3 ii) √15÷√3 = . . . . . . . . . . = . . . . . . . . . . . = . . . . . . . . . . . Rationalizing the Denominator eg: 3/√2 = . . . . . . . . = . . . . . . .. Denote in Simplest form. Eg: √44 = √4x11 = √4 x√11 = 2x√11 = 2√11 i) √40 ii) √75 iii) √54 iv) √288 v) √98 Prepared by -­‐PBP Darshana de Silva, (B.Sc, PgDBM, MBS) University of Colombo 2 Indices Exercise -­‐C (i) (169)1/2 (ii) (49y4)1/2 (iii) (25p2q2)1/2 1 (vi) 9 (iv) (216a3)1/3 (v) 2431/5 1/2 (vii) 27 1/3 (viii) (0.81)1/2 (ix) (0.125)1/3 64 Simplify (i) (27)2/3 (ii) (8)2/3 (v) (iii) (81)1/4 (vi) (0.256)0.25 (iv) (36)1/2 64 2/3 343 (vii) (16)-­‐1/2 1 11 9 (viii) (27x3)-­‐1/3 (ix) (xi) (5 -­‐4) 2 (xii) (1000) -­‐1/3 -­‐1/2 (x) (100 1/2) -­‐3 Prepared by -­‐PBP Darshana de Silva, (B.Sc, PgDBM, MBS) University of Colomb0 3 6 (xiii) (0.01) 1/2 (xiv) (41/2) 2 (xv) √a 2 b 3 3. Evaluate each of the following expressions when x = 16 and y =8. (i) 2x1/2 × y 1/3 (ii) x1/4 × y -­‐1 (iii) x -­‐2 y 4. Simplify (i) (2a)-­‐2 × 8a4 (ii) (4x1/2) × (8x3/2) (iii) 2x -­‐2 × 4x 3 (vi) (3y)-­‐1 ÷ (9y2)-­‐1 (iv) (3x)2 × 1 x2 1/2 (v) (4x1/2) ÷ x3/2 9 (vii) 10x-­‐1 y 3 × xy (viii) (3x-­‐1)-­‐2 -­‐1 5. Simplify (i) -­‐1 1/2 (ii) 1 9 3° × 2 16 1/2
1 2 2 × 4 1 81 1/4 (iii) X1/2 × ( X ½ + Y 1/2) (iv) X 2/3 ( X 4/3 + X 1/3 + X ) (v) (4X-­‐2Y)2(3X3Y-­‐4)-­‐2 Prepared by -­‐PBP Darshana de Silva, (B.Sc, PgDBM, MBS) University of Colombo 4 Exercise (Solve for x) 1. 3x = 81 2. 5x = 1/5 3. 3-­‐x = 1/27 5. 52x = 25 6. 1/4x = 64 8. 82x-­‐1 = 1/64 9. 12x = 1 11. 5x+2 + 53-­‐x = 10 (ii) log 5 125 (iii) log 100 (v) log 1003 (vi) log 3 81 1/4 4. 10x = 1/100 7. 4x+1 = 32 10. 5x+1 + 52-­‐x =53 + 1 Exercise (Evaluate) (i) log 2 64 (iv) log 3 33 Prepared by -­‐PBP Darshana de Silva, (B.Sc, PgDBM, MBS) University of Colombo 5 Logarithms 01. a=24/25 b=16/15 c=4/5 . . . . . . . . . . . loge2= logea+ logeb-­‐ 3logec . . . . . . 02. If log 2 = 0.3010, log 3 = 0.4771, log 7 = 0.8451 evaluate log (4/21) Find the value of x 03. log(x3 + 1) – log(x2 – 3x +2) = log(x2 – x + 1) 04. log e (x -­‐ 14) + log e (x -­‐ 5) = 1 05. log 10 (x2 + 1) – 2 log 10 x = 1 06. log 10 x – log x 10 = 3 07. 5(2x) = 4 1-­‐x 08. √2x+2 + 5 + √5.2 x -­‐9 = √3.2 x+2 +21 09. If x2 + y2 =7xy , prove that log(x+y) = log 3 + ½ log x + ½ log y 10. . . . . . . . . . . . . . . x√2 + y√2 = 97/9 , . . . . . . . xy = 2√2 11. 32x -­‐ 3x+1 = 4 12. 22+x + 43+x = 1020 13. 52x+1 -­‐ 25x-­‐1 = 31/25 14. 18yx -­‐ y2x = 81 , 3x = y2 15. 22+2x + 3.2x -­‐ 1 = 0 16. 2x + 3y = 44 , 17. If x = log a bc , y = log b ca and z = log c ab show that xyz = x + y + z + 2 18. 3x -­‐ 2y+2 = 10 19. log b2 a . logx2 b = ¼ logx a 20. 22x + 6.2x = 5 2x+2 + 3y+2 = 221 Find the value of x and y 2y -­‐ 3x+2 = 2 21. Find the value of X a) 22x + 2x.6= 5 b) 22x-­‐ 2x+2-­‐5=0 Find the value of X c) Solve 2 log X = log 3 + log (2X-­‐3) d) 184-­‐4x = (54√2)3x-­‐2 22. if Xp = QY2 , Show that p log X = log Q + 2 log Y or log Y=(p/2) log X-­‐(1/2) log Q Prepared by -­‐PBP Darshana de Silva, (B.Sc, PgDBM, MBS) University of Colombo 6